Functional renormalization group for the effective average action

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Functional renormalization group for the
effective average action
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physics at different length scales

• microscopic theories where the laws are
  formulated
• effective theories where observations are made
• effective theory may involve different degrees of
  freedom as compared to microscopic theory
• example the motion of the earth around the sun
  does not need an understanding of nuclear burning
  in the sun

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QCD Short and long distance degrees of freedom
are different ! Short distances quarks
and gluons Long distances baryons and
mesons How to make the transition?
confinement/chiral symmetry breaking
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collective degrees of freedom
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Hubbard model

• Electrons on a cubic lattice
• here on planes ( d 2 )
• Repulsive local interaction if two electrons are
  on the same site
• Hopping interaction between two neighboring sites

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Hubbard model
Functional integral formulation
next neighbor interaction
External parameters T temperature µ chemical
potential (doping )
U gt 0 repulsive local interaction
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In solid state physics model for everything

• Antiferromagnetism
• High Tc superconductivity
• Metal-insulator transition
• Ferromagnetism

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Antiferromagnetism in d2 Hubbard model
U/t 3
antiferro- magnetic order parameter
µ 0
Tc/t 0.115
T.Baier, E.Bick,
temperature in units of t
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antiferromagnetic order is finite size effect

• here size of experimental probe 1 cm
• vanishing order for infinite volume
• consistency with Mermin-Wagner theorem
• dependence on probe size very weak

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Collective degrees of freedom are crucial !

• for T lt Tc
• nonvanishing order parameter
• gap for fermions
• low energy excitations
• antiferromagnetic spin waves

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effective theory / microscopic theory

• sometimes only distinguished by different values
  of couplings
• sometimes different degrees of freedom
• need for methods that can cope with such
  situations

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Functional Renormalization Group

• describes flow of effective action from small to
  large length scales
• perturbative renormalization case where only
  couplings change , and couplings are small

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How to come from quarks and gluons to baryons and
mesons ?How to come from electrons to spin waves
?

• Find effective description where relevant
  degrees of freedom depend on momentum scale or
  resolution in space.
• Microscope with variable resolution
• High resolution , small piece of volume
• quarks and gluons
• Low resolution, large volume hadrons

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Wegner, Houghton
/
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effective average action
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Unified picture for scalar field theorieswith
symmetry O(N) in arbitrary dimension d and
arbitrary N

• linear or nonlinear sigma-model for
• chiral symmetry breaking in QCD
• or
• scalar model for antiferromagnetic spin waves
• (linear O(3) model )
• fermions will
  be added later

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Effective potential includes all fluctuations
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Scalar field theory
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Flow equation for average potential
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Simple one loop structure nevertheless (almost)
exact
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Infrared cutoff
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Partial differential equation for function U(k,f)
depending on two ( or more ) variables
Z k c k-?
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Regularisation

• For suitable Rk
• Momentum integral is ultraviolet and infrared
  finite
• Numerical integration possible
• Flow equation defines a regularization scheme (
  ERGE regularization )

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Integration by momentum shells

• Momentum integral
• is dominated by
• q2 k2 .
• Flow only sensitive to
• physics at scale k

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Wave function renormalization and anomalous
dimension

• for Zk (f,q2) flow equation is exact !

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Scaling form of evolution equation
On r.h.s. neither the scale k nor the wave
function renormalization Z appear
explicitly. Scaling solution no dependence on
t corresponds to second order phase transition.
Tetradis
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decoupling of heavy modes
threshold functions vanish for large w large
mass as compared to k Flow involves
effectively only modes with mass smaller or
equal k
unnecessary heavy modes are eliminated
automatically effective
theories addition of new collective modes still
needs to be done
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unified approach

• choose N
• choose d
• choose initial form of potential
• run !

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Flow of effective potential

• Ising model

CO2
Critical exponents
Experiment
T 304.15 K p 73.8.bar ? 0.442 g cm-2
S.Seide
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Critical exponents , d3
ERGE world
ERGE world
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derivative expansion

• good results already in lowest order in
  derivative expansion one function u to be
  determined
• second order derivative expansion - include field
  dependence of wave function renormalization
  three functions to be determined

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apparent convergence of derivative expansion
from talk by Bervilliers
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anomalous dimension
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Solution of partial differential equation
yields highly nontrivial non-perturbative
results despite the one loop structure
! Example Kosterlitz-Thouless phase transition
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Essential scaling d2,N2

• Flow equation contains correctly the
  non-perturbative information !
• (essential scaling usually described by vortices)

Von Gersdorff
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Kosterlitz-Thouless phase transition (d2,N2)

• Correct description of phase with
• Goldstone boson
• ( infinite correlation length )
• for TltTc

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Running renormalized d-wave superconducting order
parameter ? in doped Hubbard model
TgtTc
?
Tc
TltTc
- ln (k/?)
C.Krahl,
macroscopic scale 1 cm
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Renormalized order parameter ? and gap in
electron propagator ?in doped Hubbard model
100 ? / t
?
jump
T/Tc
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Temperature dependent anomalous dimension ?
?
T/Tc
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convergence and errors

• for precise results systematic derivative
  expansion in second order in derivatives
• includes field dependent wave function
  renormalization Z(?)
• fourth order similar results
• apparent fast convergence no series resummation
• rough error estimate by different cutoffs and
  truncations

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Effective average actionand exact
renormalization group equation
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Generating functional
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Effective average action
Loop expansion perturbation theory
with infrared cutoff in propagator
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Quantum effective action
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Exact renormalization group equation
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Proof of exact flow equation
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Truncations

• Functional differential equation
• cannot be solved exactly
• Approximative solution by truncation of
• most general form of effective action

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non-perturbative systematic expansions
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Exact flow equation for effective potential

• Evaluate exact flow equation for homogeneous
  field f .
• R.h.s. involves exact propagator in homogeneous
  background field f.

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many models have been studied along these lines

• several fields
• complicated phase structure ( e.g. 3He )
• replica trick N0
• shift in critical temperature for Bose-Einstein
  condensate with interaction ( needs resolution
  for momentum dependence of propagator )
• gauge theories

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disordered systems
Canet , Delamotte , Tissier ,
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including fermions

• no particular problem !

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Universality in ultra-cold fermionic atom gases

• with
• S. Diehl , H.Gies , J.Pawlowski

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BEC BCS crossover

• Bound molecules of two atoms
• on microscopic scale
• Bose-Einstein condensate (BEC ) for low T
• Fermions with attractive interactions
• (molecules play no role )
• BCS superfluidity at low T
• by condensation of Cooper pairs
• Crossover by Feshbach resonance
• as a transition in terms of external magnetic
  field

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chemical potential
BCS
BEC
inverse scattering length
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BEC BCS crossover

• qualitative and partially quantitative
  theoretical understanding
• mean field theory (MFT ) and first attempts beyond

concentration c a kF reduced chemical
potential s µ/eF Fermi momemtum
kF Fermi energy eF binding energy
T 0
BCS
BEC
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concentration

• c a kF , a(B) scattering length
• needs computation of density nkF3/(3p2)

dilute
dilute
dense
non- interacting Fermi gas
non- interacting Bose gas
T 0
BCS
BEC
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different methods
Quantum Monte Carlo
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QFT for non-relativistic fermions

• functional integral, action

perturbation theory Feynman rules
t euclidean time on torus with circumference
1/T s effective chemical potential
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parameters

• detuning ?(B)
• Yukawa or Feshbach coupling hf

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fermionic action

• equivalent fermionic action , in general not local

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scattering length a
a M ?/4p

• broad resonance pointlike limit
• large Feshbach coupling

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collective di-atom states

• collective degrees of freedom
• can be introduced by
• partial bosonisation
• ( Hubbard - Stratonovich transformation )

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units and dimensions

• c 1 h 1 kB 1
• momentum length-1 mass eV
• energies 2ME (momentum)2
• ( M atom mass )
• typical momentum unit Fermi momentum
• typical energy and temperature unit Fermi
  energy
• time (momentum) -2
• canonical dimensions different from relativistic
  QFT !

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rescaled action

• M drops out
• all quantities in units of kF if

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effective action

• integrate out all quantum and thermal
  fluctuations
• quantum effective action
• generates full propagators and vertices
• richer structure than classical action

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gap parameter
?
T 0
BCS
BEC
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limits
BCS for gap
condensate fraction for bosons with scattering
length 0.9 a
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temperature dependence of condensate
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condensate fraction second order phase
transition
c -1 1
free BEC
c -1 0
universal critical behavior
T/Tc
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changing degrees of freedom
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Antiferromagnetic order in the Hubbard model

• A functional renormalization group study

T.Baier, E.Bick,
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Hubbard model
Functional integral formulation
next neighbor interaction
External parameters T temperature µ chemical
potential (doping )
U gt 0 repulsive local interaction
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lattice propagator
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Fermion bilinears
Introduce sources for bilinears Functional
variation with respect to sources J yields
expectation values and correlation functions
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Partial Bosonisation

• collective bosonic variables for fermion
  bilinears
• insert identity in functional integral
• ( Hubbard-Stratonovich transformation )
• replace four fermion interaction by equivalent
  bosonic interaction ( e.g. mass and Yukawa terms)
• problem decomposition of fermion interaction
  into bilinears not unique ( Grassmann variables)

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Partially bosonised functional integral
Bosonic integration is Gaussian or solve
bosonic field equation as functional of fermion
fields and reinsert into action
equivalent to fermionic functional integral if
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fermion boson action
fermion kinetic term
boson quadratic term (classical propagator )
Yukawa coupling
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source term
is now linear in the bosonic fields
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Mean Field Theory (MFT)
Evaluate Gaussian fermionic integral in
background of bosonic field , e.g.
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Effective potential in mean field theory
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Mean field phase diagram
for two different choices of couplings same U !
Tc
Tc
µ
µ
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Mean field ambiguity
Artefact of approximation cured by inclusion
of bosonic fluctuations J.Jaeckel,
Tc
Um U? U/2
U m U/3 ,U? 0
µ
mean field phase diagram
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Rebosonization and the mean field ambiguity
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Bosonic fluctuations
boson loops
fermion loops
mean field theory
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Rebosonization

• adapt bosonization to every scale k such that
• is translated to bosonic interaction

k-dependent field redefinition
H.Gies ,
absorbs four-fermion coupling
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Modification of evolution of couplings
Evolution with k-dependent field variables
Rebosonisation
Choose ak such that no four fermion coupling is
generated
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cures mean field ambiguity
Tc
MFT
HF/SD
Flow eq.
U?/t
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conclusions

• Flow equation for effective average action
• Does it work?
• Why does it work?
• When does it work?
• How accurately does it work?

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end
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Flow equationfor theHubbard model
T.Baier , E.Bick ,
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Truncation
Concentrate on antiferromagnetism
Potential U depends only on a a2
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scale evolution of effective potential for
antiferromagnetic order parameter
boson contribution
fermion contribution
effective masses depend on a !
gap for fermions a
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running couplings
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Running mass term
unrenormalized mass term
-ln(k/t)
four-fermion interaction m-2 diverges
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dimensionless quantities
renormalized antiferromagnetic order parameter ?
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evolution of potential minimum
?
10 -2 ?
-ln(k/t)
U/t 3 , T/t 0.15
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Critical temperature
For TltTc ? remains positive for k/t gt 10-9
size of probe gt 1 cm
?
T/t0.05
T/t0.1
Tc0.115
-ln(k/t)
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Below the critical temperature
Infinite-volume-correlation-length becomes larger
than sample size finite sample finite k
order remains effectively
U 3
antiferro- magnetic order parameter
Tc/t 0.115
temperature in units of t
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Pseudocritical temperature Tpc

• Limiting temperature at which bosonic mass term
  vanishes ( ? becomes nonvanishing )
• It corresponds to a diverging four-fermion
  coupling
• This is the critical temperature computed in
  MFT !

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Pseudocritical temperature
Tpc
MFT(HF)
Flow eq.
Tc
µ
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Below the pseudocritical temperature
the reign of the goldstone bosons
effective nonlinear O(3) s - model
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critical behavior
for interval Tc lt T lt Tpc evolution as for
classical Heisenberg model cf.
Chakravarty,Halperin,Nelson
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critical correlation length

c,ß slowly varying functions exponential
growth of correlation length compatible with
observation ! at Tc correlation length reaches
sample size !
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critical behavior for order parameter and
correlation function
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Mermin-Wagner theorem ?

• No spontaneous symmetry breaking
• of continuous symmetry in d2 !

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crossover phase diagram
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shift of BEC critical temperature